We present the open-source Bayesian Atmospheric Radiative Transfer (BART) retrieval package, which produces estimates and uncertainties for an atmospheres thermal profile and chemical abundances from observations. Several BART components are also stand-alone packages, including the parallel Multi-Core Markov chain Monte Carlo (MC3), which implements several Bayesian samplers; a line-by-line radiative-transfer model, transit; a code that calculates Thermochemical Equilibrium Abundances, TEA; and a test suite for verifying radiative-transfer and retrieval codes, BARTTest. The codes are in Python and C. BART and TEA are under a Reproducible Research (RR) license, which requires reviewed-paper authors to publish a compendium of all inputs, codes, and outputs supporting the papers scientific claims. BART and TEA produce the compendiums content. Otherwise, these codes are under permissive open-source terms, as are MC3 and BARTTest, for any purpose. This paper presents an overview of the code, BARTTest, and an application to eclipse data for exoplanet HD 189733 b. Appendices address RR methodology for accelerating science, a reporting checklist for retrieval papers, the spectral resolution required for synthetic tests, and a derivation of the effective sample size required to estimate any Bayesian posterior distribution to a given precision, which determines how many iterations to run. Paper II, by Cubillos et al., presents the underlying radiative-transfer scheme and an application to transit data for exoplanet HAT-P-11b. Paper III, by Blecic et al., discusses the initialization and post-processing routines, with an application to eclipse data for exoplanet WASP-43b. We invite the community to use and improve BART and its components at http://GitHub.com/ExOSPORTS/BART/.
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